First slide

Parabola

Question

If $t_{1} + t_{2} + t_{3} = - t_{1} t_{2} t_{3}$ then the orthocentre of the triangle formed by the points $A ({at}_{1}, a (t_{1} + t_{2})) B ({at}_{2} t_{3}, a (t_{2} + t_{3})), C ({at}_{3} t_{1}, a (t_{3} + t_{1}))$ , lies on

Easy

A

x-axis
B

y-axis
C

y=x
D

x=a

Solution

Equation of altitude from the point $A ({at}_{1} t_{2}, a (t_{1} + t_{2}))$ is $y - a (t_{1} + t_{2}) = - t_{3} (x - {at}_{1} t_{2})$
Re write this equation as
$t_{3} x + y = a (t_{1} + t_{2}) + {at}_{1} t_{2} t_{3}$ …… (1)
Equation of altitude from the point $B ({at}_{2} t_{3}, a (t_{2} + t_{3}))$ is
$t_{1} x + y = a (t_{2} + t_{3}) + {at}_{1} t_{2} t_{3}$ …… (2)
The ortho center is the point of intersection of the above two lines
Hence, subtract equation (1) from (2)
$x (t_{3} - t_{1}) = a (t_{1} - t_{3}) \Rightarrow x = - a$
Substitute x=-a in the equation $t_{3} x + y = a (t_{1} + t_{2}) + {at}_{1} t_{2} t_{3}$
$\begin{matrix} t_{3} (- a) + y = a (t_{1} + t_{2}) + {at}_{1} t_{2} t_{3} \\ y = a (t_{1} + t_{2} + t_{3}) + {at}_{1} t_{2} t_{3} \\ = 0 \end{matrix}$
Therefore, the ortho center is $(- a, 0)$ it lies on x- axis.

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